// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_SELFADJOINTEIGENSOLVER_H
#define EIGEN_SELFADJOINTEIGENSOLVER_H

#include "./Tridiagonalization.h"

namespace Eigen {

template <typename _MatrixType> class GeneralizedSelfAdjointEigenSolver;

namespace internal {
    template <typename SolverType, int Size, bool IsComplex> struct direct_selfadjoint_eigenvalues;

    template <typename MatrixType, typename DiagType, typename SubDiagType>
    EIGEN_DEVICE_FUNC ComputationInfo
    computeFromTridiagonal_impl(DiagType& diag, SubDiagType& subdiag, const Index maxIterations, bool computeEigenvectors, MatrixType& eivec);
}  // namespace internal

/** \eigenvalues_module \ingroup Eigenvalues_Module
  *
  *
  * \class SelfAdjointEigenSolver
  *
  * \brief Computes eigenvalues and eigenvectors of selfadjoint matrices
  *
  * \tparam _MatrixType the type of the matrix of which we are computing the
  * eigendecomposition; this is expected to be an instantiation of the Matrix
  * class template.
  *
  * A matrix \f$ A \f$ is selfadjoint if it equals its adjoint. For real
  * matrices, this means that the matrix is symmetric: it equals its
  * transpose. This class computes the eigenvalues and eigenvectors of a
  * selfadjoint matrix. These are the scalars \f$ \lambda \f$ and vectors
  * \f$ v \f$ such that \f$ Av = \lambda v \f$.  The eigenvalues of a
  * selfadjoint matrix are always real. If \f$ D \f$ is a diagonal matrix with
  * the eigenvalues on the diagonal, and \f$ V \f$ is a matrix with the
  * eigenvectors as its columns, then \f$ A = V D V^{-1} \f$. This is called the
  * eigendecomposition.
  *
  * For a selfadjoint matrix, \f$ V \f$ is unitary, meaning its inverse is equal
  * to its adjoint, \f$ V^{-1} = V^{\dagger} \f$. If \f$ A \f$ is real, then
  * \f$ V \f$ is also real and therefore orthogonal, meaning its inverse is
  * equal to its transpose, \f$ V^{-1} = V^T \f$.
  *
  * The algorithm exploits the fact that the matrix is selfadjoint, making it
  * faster and more accurate than the general purpose eigenvalue algorithms
  * implemented in EigenSolver and ComplexEigenSolver.
  *
  * Only the \b lower \b triangular \b part of the input matrix is referenced.
  *
  * Call the function compute() to compute the eigenvalues and eigenvectors of
  * a given matrix. Alternatively, you can use the
  * SelfAdjointEigenSolver(const MatrixType&, int) constructor which computes
  * the eigenvalues and eigenvectors at construction time. Once the eigenvalue
  * and eigenvectors are computed, they can be retrieved with the eigenvalues()
  * and eigenvectors() functions.
  *
  * The documentation for SelfAdjointEigenSolver(const MatrixType&, int)
  * contains an example of the typical use of this class.
  *
  * To solve the \em generalized eigenvalue problem \f$ Av = \lambda Bv \f$ and
  * the likes, see the class GeneralizedSelfAdjointEigenSolver.
  *
  * \sa MatrixBase::eigenvalues(), class EigenSolver, class ComplexEigenSolver
  */
template <typename _MatrixType> class SelfAdjointEigenSolver
{
public:
    typedef _MatrixType MatrixType;
    enum
    {
        Size = MatrixType::RowsAtCompileTime,
        ColsAtCompileTime = MatrixType::ColsAtCompileTime,
        Options = MatrixType::Options,
        MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };

    /** \brief Scalar type for matrices of type \p _MatrixType. */
    typedef typename MatrixType::Scalar Scalar;
    typedef Eigen::Index Index;  ///< \deprecated since Eigen 3.3

    typedef Matrix<Scalar, Size, Size, ColMajor, MaxColsAtCompileTime, MaxColsAtCompileTime> EigenvectorsType;

    /** \brief Real scalar type for \p _MatrixType.
      *
      * This is just \c Scalar if #Scalar is real (e.g., \c float or
      * \c double), and the type of the real part of \c Scalar if #Scalar is
      * complex.
      */
    typedef typename NumTraits<Scalar>::Real RealScalar;

    friend struct internal::direct_selfadjoint_eigenvalues<SelfAdjointEigenSolver, Size, NumTraits<Scalar>::IsComplex>;

    /** \brief Type for vector of eigenvalues as returned by eigenvalues().
      *
      * This is a column vector with entries of type #RealScalar.
      * The length of the vector is the size of \p _MatrixType.
      */
    typedef typename internal::plain_col_type<MatrixType, RealScalar>::type RealVectorType;
    typedef Tridiagonalization<MatrixType> TridiagonalizationType;
    typedef typename TridiagonalizationType::SubDiagonalType SubDiagonalType;

    /** \brief Default constructor for fixed-size matrices.
      *
      * The default constructor is useful in cases in which the user intends to
      * perform decompositions via compute(). This constructor
      * can only be used if \p _MatrixType is a fixed-size matrix; use
      * SelfAdjointEigenSolver(Index) for dynamic-size matrices.
      *
      * Example: \include SelfAdjointEigenSolver_SelfAdjointEigenSolver.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_SelfAdjointEigenSolver.out
      */
    EIGEN_DEVICE_FUNC
    SelfAdjointEigenSolver() : m_eivec(), m_eivalues(), m_subdiag(), m_hcoeffs(), m_info(InvalidInput), m_isInitialized(false), m_eigenvectorsOk(false) {}

    /** \brief Constructor, pre-allocates memory for dynamic-size matrices.
      *
      * \param [in]  size  Positive integer, size of the matrix whose
      * eigenvalues and eigenvectors will be computed.
      *
      * This constructor is useful for dynamic-size matrices, when the user
      * intends to perform decompositions via compute(). The \p size
      * parameter is only used as a hint. It is not an error to give a wrong
      * \p size, but it may impair performance.
      *
      * \sa compute() for an example
      */
    EIGEN_DEVICE_FUNC
    explicit SelfAdjointEigenSolver(Index size)
        : m_eivec(size, size), m_eivalues(size), m_subdiag(size > 1 ? size - 1 : 1), m_hcoeffs(size > 1 ? size - 1 : 1), m_isInitialized(false),
          m_eigenvectorsOk(false)
    {
    }

    /** \brief Constructor; computes eigendecomposition of given matrix.
      *
      * \param[in]  matrix  Selfadjoint matrix whose eigendecomposition is to
      *    be computed. Only the lower triangular part of the matrix is referenced.
      * \param[in]  options Can be #ComputeEigenvectors (default) or #EigenvaluesOnly.
      *
      * This constructor calls compute(const MatrixType&, int) to compute the
      * eigenvalues of the matrix \p matrix. The eigenvectors are computed if
      * \p options equals #ComputeEigenvectors.
      *
      * Example: \include SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType.out
      *
      * \sa compute(const MatrixType&, int)
      */
    template <typename InputType>
    EIGEN_DEVICE_FUNC explicit SelfAdjointEigenSolver(const EigenBase<InputType>& matrix, int options = ComputeEigenvectors)
        : m_eivec(matrix.rows(), matrix.cols()), m_eivalues(matrix.cols()), m_subdiag(matrix.rows() > 1 ? matrix.rows() - 1 : 1),
          m_hcoeffs(matrix.cols() > 1 ? matrix.cols() - 1 : 1), m_isInitialized(false), m_eigenvectorsOk(false)
    {
        compute(matrix.derived(), options);
    }

    /** \brief Computes eigendecomposition of given matrix.
      *
      * \param[in]  matrix  Selfadjoint matrix whose eigendecomposition is to
      *    be computed. Only the lower triangular part of the matrix is referenced.
      * \param[in]  options Can be #ComputeEigenvectors (default) or #EigenvaluesOnly.
      * \returns    Reference to \c *this
      *
      * This function computes the eigenvalues of \p matrix.  The eigenvalues()
      * function can be used to retrieve them.  If \p options equals #ComputeEigenvectors,
      * then the eigenvectors are also computed and can be retrieved by
      * calling eigenvectors().
      *
      * This implementation uses a symmetric QR algorithm. The matrix is first
      * reduced to tridiagonal form using the Tridiagonalization class. The
      * tridiagonal matrix is then brought to diagonal form with implicit
      * symmetric QR steps with Wilkinson shift. Details can be found in
      * Section 8.3 of Golub \& Van Loan, <i>%Matrix Computations</i>.
      *
      * The cost of the computation is about \f$ 9n^3 \f$ if the eigenvectors
      * are required and \f$ 4n^3/3 \f$ if they are not required.
      *
      * This method reuses the memory in the SelfAdjointEigenSolver object that
      * was allocated when the object was constructed, if the size of the
      * matrix does not change.
      *
      * Example: \include SelfAdjointEigenSolver_compute_MatrixType.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_compute_MatrixType.out
      *
      * \sa SelfAdjointEigenSolver(const MatrixType&, int)
      */
    template <typename InputType> EIGEN_DEVICE_FUNC SelfAdjointEigenSolver& compute(const EigenBase<InputType>& matrix, int options = ComputeEigenvectors);

    /** \brief Computes eigendecomposition of given matrix using a closed-form algorithm
      *
      * This is a variant of compute(const MatrixType&, int options) which
      * directly solves the underlying polynomial equation.
      * 
      * Currently only 2x2 and 3x3 matrices for which the sizes are known at compile time are supported (e.g., Matrix3d).
      * 
      * This method is usually significantly faster than the QR iterative algorithm
      * but it might also be less accurate. It is also worth noting that
      * for 3x3 matrices it involves trigonometric operations which are
      * not necessarily available for all scalar types.
      * 
      * For the 3x3 case, we observed the following worst case relative error regarding the eigenvalues:
      *   - double: 1e-8
      *   - float:  1e-3
      *
      * \sa compute(const MatrixType&, int options)
      */
    EIGEN_DEVICE_FUNC
    SelfAdjointEigenSolver& computeDirect(const MatrixType& matrix, int options = ComputeEigenvectors);

    /**
      *\brief Computes the eigen decomposition from a tridiagonal symmetric matrix
      *
      * \param[in] diag The vector containing the diagonal of the matrix.
      * \param[in] subdiag The subdiagonal of the matrix.
      * \param[in] options Can be #ComputeEigenvectors (default) or #EigenvaluesOnly.
      * \returns Reference to \c *this
      *
      * This function assumes that the matrix has been reduced to tridiagonal form.
      *
      * \sa compute(const MatrixType&, int) for more information
      */
    SelfAdjointEigenSolver& computeFromTridiagonal(const RealVectorType& diag, const SubDiagonalType& subdiag, int options = ComputeEigenvectors);

    /** \brief Returns the eigenvectors of given matrix.
      *
      * \returns  A const reference to the matrix whose columns are the eigenvectors.
      *
      * \pre The eigenvectors have been computed before.
      *
      * Column \f$ k \f$ of the returned matrix is an eigenvector corresponding
      * to eigenvalue number \f$ k \f$ as returned by eigenvalues().  The
      * eigenvectors are normalized to have (Euclidean) norm equal to one. If
      * this object was used to solve the eigenproblem for the selfadjoint
      * matrix \f$ A \f$, then the matrix returned by this function is the
      * matrix \f$ V \f$ in the eigendecomposition \f$ A = V D V^{-1} \f$.
      *
      * For a selfadjoint matrix, \f$ V \f$ is unitary, meaning its inverse is equal
      * to its adjoint, \f$ V^{-1} = V^{\dagger} \f$. If \f$ A \f$ is real, then
      * \f$ V \f$ is also real and therefore orthogonal, meaning its inverse is
      * equal to its transpose, \f$ V^{-1} = V^T \f$.
      *
      * Example: \include SelfAdjointEigenSolver_eigenvectors.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_eigenvectors.out
      *
      * \sa eigenvalues()
      */
    EIGEN_DEVICE_FUNC
    const EigenvectorsType& eigenvectors() const
    {
        eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
        eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
        return m_eivec;
    }

    /** \brief Returns the eigenvalues of given matrix.
      *
      * \returns A const reference to the column vector containing the eigenvalues.
      *
      * \pre The eigenvalues have been computed before.
      *
      * The eigenvalues are repeated according to their algebraic multiplicity,
      * so there are as many eigenvalues as rows in the matrix. The eigenvalues
      * are sorted in increasing order.
      *
      * Example: \include SelfAdjointEigenSolver_eigenvalues.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_eigenvalues.out
      *
      * \sa eigenvectors(), MatrixBase::eigenvalues()
      */
    EIGEN_DEVICE_FUNC
    const RealVectorType& eigenvalues() const
    {
        eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
        return m_eivalues;
    }

    /** \brief Computes the positive-definite square root of the matrix.
      *
      * \returns the positive-definite square root of the matrix
      *
      * \pre The eigenvalues and eigenvectors of a positive-definite matrix
      * have been computed before.
      *
      * The square root of a positive-definite matrix \f$ A \f$ is the
      * positive-definite matrix whose square equals \f$ A \f$. This function
      * uses the eigendecomposition \f$ A = V D V^{-1} \f$ to compute the
      * square root as \f$ A^{1/2} = V D^{1/2} V^{-1} \f$.
      *
      * Example: \include SelfAdjointEigenSolver_operatorSqrt.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_operatorSqrt.out
      *
      * \sa operatorInverseSqrt(), <a href="unsupported/group__MatrixFunctions__Module.html">MatrixFunctions Module</a>
      */
    EIGEN_DEVICE_FUNC
    MatrixType operatorSqrt() const
    {
        eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
        eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
        return m_eivec * m_eivalues.cwiseSqrt().asDiagonal() * m_eivec.adjoint();
    }

    /** \brief Computes the inverse square root of the matrix.
      *
      * \returns the inverse positive-definite square root of the matrix
      *
      * \pre The eigenvalues and eigenvectors of a positive-definite matrix
      * have been computed before.
      *
      * This function uses the eigendecomposition \f$ A = V D V^{-1} \f$ to
      * compute the inverse square root as \f$ V D^{-1/2} V^{-1} \f$. This is
      * cheaper than first computing the square root with operatorSqrt() and
      * then its inverse with MatrixBase::inverse().
      *
      * Example: \include SelfAdjointEigenSolver_operatorInverseSqrt.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_operatorInverseSqrt.out
      *
      * \sa operatorSqrt(), MatrixBase::inverse(), <a href="unsupported/group__MatrixFunctions__Module.html">MatrixFunctions Module</a>
      */
    EIGEN_DEVICE_FUNC
    MatrixType operatorInverseSqrt() const
    {
        eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
        eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
        return m_eivec * m_eivalues.cwiseInverse().cwiseSqrt().asDiagonal() * m_eivec.adjoint();
    }

    /** \brief Reports whether previous computation was successful.
      *
      * \returns \c Success if computation was successful, \c NoConvergence otherwise.
      */
    EIGEN_DEVICE_FUNC
    ComputationInfo info() const
    {
        eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
        return m_info;
    }

    /** \brief Maximum number of iterations.
      *
      * The algorithm terminates if it does not converge within m_maxIterations * n iterations, where n
      * denotes the size of the matrix. This value is currently set to 30 (copied from LAPACK).
      */
    static const int m_maxIterations = 30;

protected:
    static EIGEN_DEVICE_FUNC void check_template_parameters() { EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); }

    EigenvectorsType m_eivec;
    RealVectorType m_eivalues;
    typename TridiagonalizationType::SubDiagonalType m_subdiag;
    typename TridiagonalizationType::CoeffVectorType m_hcoeffs;
    ComputationInfo m_info;
    bool m_isInitialized;
    bool m_eigenvectorsOk;
};

namespace internal {
    /** \internal
  *
  * \eigenvalues_module \ingroup Eigenvalues_Module
  *
  * Performs a QR step on a tridiagonal symmetric matrix represented as a
  * pair of two vectors \a diag and \a subdiag.
  *
  * \param diag the diagonal part of the input selfadjoint tridiagonal matrix
  * \param subdiag the sub-diagonal part of the input selfadjoint tridiagonal matrix
  * \param start starting index of the submatrix to work on
  * \param end last+1 index of the submatrix to work on
  * \param matrixQ pointer to the column-major matrix holding the eigenvectors, can be 0
  * \param n size of the input matrix
  *
  * For compilation efficiency reasons, this procedure does not use eigen expression
  * for its arguments.
  *
  * Implemented from Golub's "Matrix Computations", algorithm 8.3.2:
  * "implicit symmetric QR step with Wilkinson shift"
  */
    template <int StorageOrder, typename RealScalar, typename Scalar, typename Index>
    EIGEN_DEVICE_FUNC static void tridiagonal_qr_step(RealScalar* diag, RealScalar* subdiag, Index start, Index end, Scalar* matrixQ, Index n);
}  // namespace internal

template <typename MatrixType>
template <typename InputType>
EIGEN_DEVICE_FUNC SelfAdjointEigenSolver<MatrixType>& SelfAdjointEigenSolver<MatrixType>::compute(const EigenBase<InputType>& a_matrix, int options)
{
    check_template_parameters();

    const InputType& matrix(a_matrix.derived());

    EIGEN_USING_STD(abs);
    eigen_assert(matrix.cols() == matrix.rows());
    eigen_assert((options & ~(EigVecMask | GenEigMask)) == 0 && (options & EigVecMask) != EigVecMask && "invalid option parameter");
    bool computeEigenvectors = (options & ComputeEigenvectors) == ComputeEigenvectors;
    Index n = matrix.cols();
    m_eivalues.resize(n, 1);

    if (n == 1)
    {
        m_eivec = matrix;
        m_eivalues.coeffRef(0, 0) = numext::real(m_eivec.coeff(0, 0));
        if (computeEigenvectors)
            m_eivec.setOnes(n, n);
        m_info = Success;
        m_isInitialized = true;
        m_eigenvectorsOk = computeEigenvectors;
        return *this;
    }

    // declare some aliases
    RealVectorType& diag = m_eivalues;
    EigenvectorsType& mat = m_eivec;

    // map the matrix coefficients to [-1:1] to avoid over- and underflow.
    mat = matrix.template triangularView<Lower>();
    RealScalar scale = mat.cwiseAbs().maxCoeff();
    if (scale == RealScalar(0))
        scale = RealScalar(1);
    mat.template triangularView<Lower>() /= scale;
    m_subdiag.resize(n - 1);
    m_hcoeffs.resize(n - 1);
    internal::tridiagonalization_inplace(mat, diag, m_subdiag, m_hcoeffs, computeEigenvectors);

    m_info = internal::computeFromTridiagonal_impl(diag, m_subdiag, m_maxIterations, computeEigenvectors, m_eivec);

    // scale back the eigen values
    m_eivalues *= scale;

    m_isInitialized = true;
    m_eigenvectorsOk = computeEigenvectors;
    return *this;
}

template <typename MatrixType>
SelfAdjointEigenSolver<MatrixType>&
SelfAdjointEigenSolver<MatrixType>::computeFromTridiagonal(const RealVectorType& diag, const SubDiagonalType& subdiag, int options)
{
    //TODO : Add an option to scale the values beforehand
    bool computeEigenvectors = (options & ComputeEigenvectors) == ComputeEigenvectors;

    m_eivalues = diag;
    m_subdiag = subdiag;
    if (computeEigenvectors)
    {
        m_eivec.setIdentity(diag.size(), diag.size());
    }
    m_info = internal::computeFromTridiagonal_impl(m_eivalues, m_subdiag, m_maxIterations, computeEigenvectors, m_eivec);

    m_isInitialized = true;
    m_eigenvectorsOk = computeEigenvectors;
    return *this;
}

namespace internal {
    /**
  * \internal
  * \brief Compute the eigendecomposition from a tridiagonal matrix
  *
  * \param[in,out] diag : On input, the diagonal of the matrix, on output the eigenvalues
  * \param[in,out] subdiag : The subdiagonal part of the matrix (entries are modified during the decomposition)
  * \param[in] maxIterations : the maximum number of iterations
  * \param[in] computeEigenvectors : whether the eigenvectors have to be computed or not
  * \param[out] eivec : The matrix to store the eigenvectors if computeEigenvectors==true. Must be allocated on input.
  * \returns \c Success or \c NoConvergence
  */
    template <typename MatrixType, typename DiagType, typename SubDiagType>
    EIGEN_DEVICE_FUNC ComputationInfo
    computeFromTridiagonal_impl(DiagType& diag, SubDiagType& subdiag, const Index maxIterations, bool computeEigenvectors, MatrixType& eivec)
    {
        ComputationInfo info;
        typedef typename MatrixType::Scalar Scalar;

        Index n = diag.size();
        Index end = n - 1;
        Index start = 0;
        Index iter = 0;  // total number of iterations

        typedef typename DiagType::RealScalar RealScalar;
        const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
        const RealScalar precision_inv = RealScalar(1) / NumTraits<RealScalar>::epsilon();
        while (end > 0)
        {
            for (Index i = start; i < end; ++i)
            {
                if (numext::abs(subdiag[i]) < considerAsZero)
                {
                    subdiag[i] = RealScalar(0);
                }
                else
                {
                    // abs(subdiag[i]) <= epsilon * sqrt(abs(diag[i]) + abs(diag[i+1]))
                    // Scaled to prevent underflows.
                    const RealScalar scaled_subdiag = precision_inv * subdiag[i];
                    if (scaled_subdiag * scaled_subdiag <= (numext::abs(diag[i]) + numext::abs(diag[i + 1])))
                    {
                        subdiag[i] = RealScalar(0);
                    }
                }
            }

            // find the largest unreduced block at the end of the matrix.
            while (end > 0 && subdiag[end - 1] == RealScalar(0)) { end--; }
            if (end <= 0)
                break;

            // if we spent too many iterations, we give up
            iter++;
            if (iter > maxIterations * n)
                break;

            start = end - 1;
            while (start > 0 && subdiag[start - 1] != 0) start--;

            internal::tridiagonal_qr_step<MatrixType::Flags & RowMajorBit ? RowMajor : ColMajor>(
                diag.data(), subdiag.data(), start, end, computeEigenvectors ? eivec.data() : (Scalar*)0, n);
        }
        if (iter <= maxIterations * n)
            info = Success;
        else
            info = NoConvergence;

        // Sort eigenvalues and corresponding vectors.
        // TODO make the sort optional ?
        // TODO use a better sort algorithm !!
        if (info == Success)
        {
            for (Index i = 0; i < n - 1; ++i)
            {
                Index k;
                diag.segment(i, n - i).minCoeff(&k);
                if (k > 0)
                {
                    numext::swap(diag[i], diag[k + i]);
                    if (computeEigenvectors)
                        eivec.col(i).swap(eivec.col(k + i));
                }
            }
        }
        return info;
    }

    template <typename SolverType, int Size, bool IsComplex> struct direct_selfadjoint_eigenvalues
    {
        EIGEN_DEVICE_FUNC
        static inline void run(SolverType& eig, const typename SolverType::MatrixType& A, int options) { eig.compute(A, options); }
    };

    template <typename SolverType> struct direct_selfadjoint_eigenvalues<SolverType, 3, false>
    {
        typedef typename SolverType::MatrixType MatrixType;
        typedef typename SolverType::RealVectorType VectorType;
        typedef typename SolverType::Scalar Scalar;
        typedef typename SolverType::EigenvectorsType EigenvectorsType;

        /** \internal
   * Computes the roots of the characteristic polynomial of \a m.
   * For numerical stability m.trace() should be near zero and to avoid over- or underflow m should be normalized.
   */
        EIGEN_DEVICE_FUNC
        static inline void computeRoots(const MatrixType& m, VectorType& roots)
        {
            EIGEN_USING_STD(sqrt)
            EIGEN_USING_STD(atan2)
            EIGEN_USING_STD(cos)
            EIGEN_USING_STD(sin)
            const Scalar s_inv3 = Scalar(1) / Scalar(3);
            const Scalar s_sqrt3 = sqrt(Scalar(3));

            // The characteristic equation is x^3 - c2*x^2 + c1*x - c0 = 0.  The
            // eigenvalues are the roots to this equation, all guaranteed to be
            // real-valued, because the matrix is symmetric.
            Scalar c0 = m(0, 0) * m(1, 1) * m(2, 2) + Scalar(2) * m(1, 0) * m(2, 0) * m(2, 1) - m(0, 0) * m(2, 1) * m(2, 1) - m(1, 1) * m(2, 0) * m(2, 0) -
                        m(2, 2) * m(1, 0) * m(1, 0);
            Scalar c1 = m(0, 0) * m(1, 1) - m(1, 0) * m(1, 0) + m(0, 0) * m(2, 2) - m(2, 0) * m(2, 0) + m(1, 1) * m(2, 2) - m(2, 1) * m(2, 1);
            Scalar c2 = m(0, 0) + m(1, 1) + m(2, 2);

            // Construct the parameters used in classifying the roots of the equation
            // and in solving the equation for the roots in closed form.
            Scalar c2_over_3 = c2 * s_inv3;
            Scalar a_over_3 = (c2 * c2_over_3 - c1) * s_inv3;
            a_over_3 = numext::maxi(a_over_3, Scalar(0));

            Scalar half_b = Scalar(0.5) * (c0 + c2_over_3 * (Scalar(2) * c2_over_3 * c2_over_3 - c1));

            Scalar q = a_over_3 * a_over_3 * a_over_3 - half_b * half_b;
            q = numext::maxi(q, Scalar(0));

            // Compute the eigenvalues by solving for the roots of the polynomial.
            Scalar rho = sqrt(a_over_3);
            Scalar theta = atan2(sqrt(q), half_b) * s_inv3;  // since sqrt(q) > 0, atan2 is in [0, pi] and theta is in [0, pi/3]
            Scalar cos_theta = cos(theta);
            Scalar sin_theta = sin(theta);
            // roots are already sorted, since cos is monotonically decreasing on [0, pi]
            roots(0) = c2_over_3 - rho * (cos_theta + s_sqrt3 * sin_theta);  // == 2*rho*cos(theta+2pi/3)
            roots(1) = c2_over_3 - rho * (cos_theta - s_sqrt3 * sin_theta);  // == 2*rho*cos(theta+ pi/3)
            roots(2) = c2_over_3 + Scalar(2) * rho * cos_theta;
        }

        EIGEN_DEVICE_FUNC
        static inline bool extract_kernel(MatrixType& mat, Ref<VectorType> res, Ref<VectorType> representative)
        {
            EIGEN_USING_STD(abs);
            EIGEN_USING_STD(sqrt);
            Index i0;
            // Find non-zero column i0 (by construction, there must exist a non zero coefficient on the diagonal):
            mat.diagonal().cwiseAbs().maxCoeff(&i0);
            // mat.col(i0) is a good candidate for an orthogonal vector to the current eigenvector,
            // so let's save it:
            representative = mat.col(i0);
            Scalar n0, n1;
            VectorType c0, c1;
            n0 = (c0 = representative.cross(mat.col((i0 + 1) % 3))).squaredNorm();
            n1 = (c1 = representative.cross(mat.col((i0 + 2) % 3))).squaredNorm();
            if (n0 > n1)
                res = c0 / sqrt(n0);
            else
                res = c1 / sqrt(n1);

            return true;
        }

        EIGEN_DEVICE_FUNC
        static inline void run(SolverType& solver, const MatrixType& mat, int options)
        {
            eigen_assert(mat.cols() == 3 && mat.cols() == mat.rows());
            eigen_assert((options & ~(EigVecMask | GenEigMask)) == 0 && (options & EigVecMask) != EigVecMask && "invalid option parameter");
            bool computeEigenvectors = (options & ComputeEigenvectors) == ComputeEigenvectors;

            EigenvectorsType& eivecs = solver.m_eivec;
            VectorType& eivals = solver.m_eivalues;

            // Shift the matrix to the mean eigenvalue and map the matrix coefficients to [-1:1] to avoid over- and underflow.
            Scalar shift = mat.trace() / Scalar(3);
            // TODO Avoid this copy. Currently it is necessary to suppress bogus values when determining maxCoeff and for computing the eigenvectors later
            MatrixType scaledMat = mat.template selfadjointView<Lower>();
            scaledMat.diagonal().array() -= shift;
            Scalar scale = scaledMat.cwiseAbs().maxCoeff();
            if (scale > 0)
                scaledMat /= scale;  // TODO for scale==0 we could save the remaining operations

            // compute the eigenvalues
            computeRoots(scaledMat, eivals);

            // compute the eigenvectors
            if (computeEigenvectors)
            {
                if ((eivals(2) - eivals(0)) <= Eigen::NumTraits<Scalar>::epsilon())
                {
                    // All three eigenvalues are numerically the same
                    eivecs.setIdentity();
                }
                else
                {
                    MatrixType tmp;
                    tmp = scaledMat;

                    // Compute the eigenvector of the most distinct eigenvalue
                    Scalar d0 = eivals(2) - eivals(1);
                    Scalar d1 = eivals(1) - eivals(0);
                    Index k(0), l(2);
                    if (d0 > d1)
                    {
                        numext::swap(k, l);
                        d0 = d1;
                    }

                    // Compute the eigenvector of index k
                    {
                        tmp.diagonal().array() -= eivals(k);
                        // By construction, 'tmp' is of rank 2, and its kernel corresponds to the respective eigenvector.
                        extract_kernel(tmp, eivecs.col(k), eivecs.col(l));
                    }

                    // Compute eigenvector of index l
                    if (d0 <= 2 * Eigen::NumTraits<Scalar>::epsilon() * d1)
                    {
                        // If d0 is too small, then the two other eigenvalues are numerically the same,
                        // and thus we only have to ortho-normalize the near orthogonal vector we saved above.
                        eivecs.col(l) -= eivecs.col(k).dot(eivecs.col(l)) * eivecs.col(l);
                        eivecs.col(l).normalize();
                    }
                    else
                    {
                        tmp = scaledMat;
                        tmp.diagonal().array() -= eivals(l);

                        VectorType dummy;
                        extract_kernel(tmp, eivecs.col(l), dummy);
                    }

                    // Compute last eigenvector from the other two
                    eivecs.col(1) = eivecs.col(2).cross(eivecs.col(0)).normalized();
                }
            }

            // Rescale back to the original size.
            eivals *= scale;
            eivals.array() += shift;

            solver.m_info = Success;
            solver.m_isInitialized = true;
            solver.m_eigenvectorsOk = computeEigenvectors;
        }
    };

    // 2x2 direct eigenvalues decomposition, code from Hauke Heibel
    template <typename SolverType> struct direct_selfadjoint_eigenvalues<SolverType, 2, false>
    {
        typedef typename SolverType::MatrixType MatrixType;
        typedef typename SolverType::RealVectorType VectorType;
        typedef typename SolverType::Scalar Scalar;
        typedef typename SolverType::EigenvectorsType EigenvectorsType;

        EIGEN_DEVICE_FUNC
        static inline void computeRoots(const MatrixType& m, VectorType& roots)
        {
            EIGEN_USING_STD(sqrt);
            const Scalar t0 = Scalar(0.5) * sqrt(numext::abs2(m(0, 0) - m(1, 1)) + Scalar(4) * numext::abs2(m(1, 0)));
            const Scalar t1 = Scalar(0.5) * (m(0, 0) + m(1, 1));
            roots(0) = t1 - t0;
            roots(1) = t1 + t0;
        }

        EIGEN_DEVICE_FUNC
        static inline void run(SolverType& solver, const MatrixType& mat, int options)
        {
            EIGEN_USING_STD(sqrt);
            EIGEN_USING_STD(abs);

            eigen_assert(mat.cols() == 2 && mat.cols() == mat.rows());
            eigen_assert((options & ~(EigVecMask | GenEigMask)) == 0 && (options & EigVecMask) != EigVecMask && "invalid option parameter");
            bool computeEigenvectors = (options & ComputeEigenvectors) == ComputeEigenvectors;

            EigenvectorsType& eivecs = solver.m_eivec;
            VectorType& eivals = solver.m_eivalues;

            // Shift the matrix to the mean eigenvalue and map the matrix coefficients to [-1:1] to avoid over- and underflow.
            Scalar shift = mat.trace() / Scalar(2);
            MatrixType scaledMat = mat;
            scaledMat.coeffRef(0, 1) = mat.coeff(1, 0);
            scaledMat.diagonal().array() -= shift;
            Scalar scale = scaledMat.cwiseAbs().maxCoeff();
            if (scale > Scalar(0))
                scaledMat /= scale;

            // Compute the eigenvalues
            computeRoots(scaledMat, eivals);

            // compute the eigen vectors
            if (computeEigenvectors)
            {
                if ((eivals(1) - eivals(0)) <= abs(eivals(1)) * Eigen::NumTraits<Scalar>::epsilon())
                {
                    eivecs.setIdentity();
                }
                else
                {
                    scaledMat.diagonal().array() -= eivals(1);
                    Scalar a2 = numext::abs2(scaledMat(0, 0));
                    Scalar c2 = numext::abs2(scaledMat(1, 1));
                    Scalar b2 = numext::abs2(scaledMat(1, 0));
                    if (a2 > c2)
                    {
                        eivecs.col(1) << -scaledMat(1, 0), scaledMat(0, 0);
                        eivecs.col(1) /= sqrt(a2 + b2);
                    }
                    else
                    {
                        eivecs.col(1) << -scaledMat(1, 1), scaledMat(1, 0);
                        eivecs.col(1) /= sqrt(c2 + b2);
                    }

                    eivecs.col(0) << eivecs.col(1).unitOrthogonal();
                }
            }

            // Rescale back to the original size.
            eivals *= scale;
            eivals.array() += shift;

            solver.m_info = Success;
            solver.m_isInitialized = true;
            solver.m_eigenvectorsOk = computeEigenvectors;
        }
    };

}  // namespace internal

template <typename MatrixType>
EIGEN_DEVICE_FUNC SelfAdjointEigenSolver<MatrixType>& SelfAdjointEigenSolver<MatrixType>::computeDirect(const MatrixType& matrix, int options)
{
    internal::direct_selfadjoint_eigenvalues<SelfAdjointEigenSolver, Size, NumTraits<Scalar>::IsComplex>::run(*this, matrix, options);
    return *this;
}

namespace internal {

    // Francis implicit QR step.
    template <int StorageOrder, typename RealScalar, typename Scalar, typename Index>
    EIGEN_DEVICE_FUNC static void tridiagonal_qr_step(RealScalar* diag, RealScalar* subdiag, Index start, Index end, Scalar* matrixQ, Index n)
    {
        // Wilkinson Shift.
        RealScalar td = (diag[end - 1] - diag[end]) * RealScalar(0.5);
        RealScalar e = subdiag[end - 1];
        // Note that thanks to scaling, e^2 or td^2 cannot overflow, however they can still
        // underflow thus leading to inf/NaN values when using the following commented code:
        //   RealScalar e2 = numext::abs2(subdiag[end-1]);
        //   RealScalar mu = diag[end] - e2 / (td + (td>0 ? 1 : -1) * sqrt(td*td + e2));
        // This explain the following, somewhat more complicated, version:
        RealScalar mu = diag[end];
        if (td == RealScalar(0))
        {
            mu -= numext::abs(e);
        }
        else if (e != RealScalar(0))
        {
            const RealScalar e2 = numext::abs2(e);
            const RealScalar h = numext::hypot(td, e);
            if (e2 == RealScalar(0))
            {
                mu -= e / ((td + (td > RealScalar(0) ? h : -h)) / e);
            }
            else
            {
                mu -= e2 / (td + (td > RealScalar(0) ? h : -h));
            }
        }

        RealScalar x = diag[start] - mu;
        RealScalar z = subdiag[start];
        // If z ever becomes zero, the Givens rotation will be the identity and
        // z will stay zero for all future iterations.
        for (Index k = start; k < end && z != RealScalar(0); ++k)
        {
            JacobiRotation<RealScalar> rot;
            rot.makeGivens(x, z);

            // do T = G' T G
            RealScalar sdk = rot.s() * diag[k] + rot.c() * subdiag[k];
            RealScalar dkp1 = rot.s() * subdiag[k] + rot.c() * diag[k + 1];

            diag[k] = rot.c() * (rot.c() * diag[k] - rot.s() * subdiag[k]) - rot.s() * (rot.c() * subdiag[k] - rot.s() * diag[k + 1]);
            diag[k + 1] = rot.s() * sdk + rot.c() * dkp1;
            subdiag[k] = rot.c() * sdk - rot.s() * dkp1;

            if (k > start)
                subdiag[k - 1] = rot.c() * subdiag[k - 1] - rot.s() * z;

            // "Chasing the bulge" to return to triangular form.
            x = subdiag[k];
            if (k < end - 1)
            {
                z = -rot.s() * subdiag[k + 1];
                subdiag[k + 1] = rot.c() * subdiag[k + 1];
            }

            // apply the givens rotation to the unit matrix Q = Q * G
            if (matrixQ)
            {
                // FIXME if StorageOrder == RowMajor this operation is not very efficient
                Map<Matrix<Scalar, Dynamic, Dynamic, StorageOrder>> q(matrixQ, n, n);
                q.applyOnTheRight(k, k + 1, rot);
            }
        }
    }

}  // end namespace internal

}  // end namespace Eigen

#endif  // EIGEN_SELFADJOINTEIGENSOLVER_H
